Welcome to MM207 - Statistics!

Unit 6 Seminar:Inferential Statistics and

Confidence Intervals

Definition ReviewPopulation - a set of measurements

Parameters described the characteristics of a population.

Sample: a subset of measurements from the populationStatistics describe the characteristics of a sample.

Most of the time we do not have the entire population, we have a sample from the population.

Therefore, we must use sample statistics to estimate population parameters.

We use a confidence interval to estimate a population mean or a proportion.

Confidence Intervals for μ or p

There are two steps

1. Find E (MoE or margin of error).

2. Find the interval.

Critical Formula σ Known

• Margin of Error [E]• E = zc (σ/√n); the level of

confidence is determined by the value selected for zc

• C-Confidence Interval• xbar – E < µ < xbar + E

• Minimum Sample Size• N = (zc*σ/E)^2

Step 2: Compute the Interval

The interval has a lower number and an upper number

For estimating μ

xbar – E < μ < xbar + E

For estimating p

phat – E < p < phat + E

Example 1: CI for μ, n ≥ 30n = 40xbar = 12σ = 5

Find the 95% CI for μ.

Step 1: Find E Step 2: Find the interval Since n ≥ 30, σ known xbar – E < μ < xbar + E E = zc * σ / √[n] 12 – 1.55 < μ < 12 + 1.55 E = 1.96 * 5 / √[40] 10.45 < μ < 13.55 E = 9.8 / 6.32455532 E ≈ 1.549516054 ≈ 1.55

Use the t-table, the bottom row, to find zc = 1.96Or use CONFIDENCE in Excel to find E

Excel for Confidence IntervalSigma Known (z)

Alpha is the complement of the confidence interval, this is for the 80% confidence interval

This is E

Critical Formula Small Samples• t-Distribution

– t = [xbar - µ] / s/√n• Margin of Error [E]

– E = tc (s/√n); the level of confidence is determined by the value selected for zc

• C-Confidence Interval– xbar – E < µ < xbar + E

• Minimum Sample Size– N = (tc*s/E)^2

Example 2: CI for μ, n < 30n = 20df = 19xbar = 12s = 5

Find the 95% CI for μ.

Step 1: Find E Step 2: Find the interval n < 30, σ not known xbar – E < μ < xbar + E df = 19 12 – 2.34 < μ < 12 + 2.34

E = tc * s / √[n] 9.66 < μ < 14.34 E = 2.093 * 5 / √[20] E = 10.465 / 4.472135955

E ≈ 2.340045138 ≈ 2.34Use the t-table, df = 19, to find 2.093

Excel for Confidence IntervalSmall Samples (t)

Score     Mean 13Standard Error 1.724819Median 11.5Mode 6Standard Deviation 6.899275Sample Variance 47.6Kurtosis -0.21368Skewness 0.77674Range 23Minimum 5Maximum 28Sum 208Count 16Confidence Level(99.0%) 5.082546

This is the function that will give you E using the t distribution

E

z-Estimate of a Proportion

• Sample proportion 0.3333• Sample size 300• Confidence level 0.99

• Confidence Interval Estimate 0.3333 +/- 0.0701

• Lower confidence limit 0.2632

• Upper confidence limit 0.4034

This is a home grown procedure. Enter the data on the left.The answers will be shown in Red.

Example 3: CI for p n = 400phat = 0.6, qhat = 1 – 0.6 = 0.4

Find the 95% CI for p.

nphat = 240 > 5, nqhat = 160 > 5, ok to use zc

Step 1: Find E Step 2: Find the intervalE = zc * √[pq / n] phat – E < p < phat + EE = 1.96 * √ [(0.6 * 0.4) / 400] 0.6 – 0.048 < p < 0.6 + 0.048E = 1.96 * √ [0.24 / 400] 0.552 < p < 0.648E = 1.96 * .024494897

E ≈ 0.048009998 ≈ 0.048

Example 4: Choosing the Normal or t-DistributionPage 329, using the flow chart

n = 25σ = $28,000xbar = $181,000

Normal or t-Distribution (zc or tc )?

n = 18s = $24,000xbar = $162,000

Normal or t-Distribution?

Other Topics

• Finding a minimum sample size for a confidence interval

• Finding zc for a confidence level• Interpreting a confidence interval• Comparing confidence intervals for a level of

90%, 95%, and 99%

Finding a minimum sample size for a confidence interval

Page 316

Find n for a 99% CI given σ ≈ s ≈ 10 and E = 3.2

n = [(zc * σ) / E]2

n = [2.575* 10 / 3.2]2

n = [25.75 / 3.2]2

n = [8.046875]2

n = 64.75 or 65

Note: Always round up! For example, you would round 72.1 to 73 because we need at least 72.1 for the sample size.

Finding Zc for a Confidence Level

Sometimes the zc for the confidence level is not provided in a table.

Find the zc for an 85% CI. This zc is not in the t-table.

1/2(1 - 0.85) = 0.15/2 = 0.075

Find the z for 0.0750 in the Standard Normal Table

zc = - 1.44 or zc = 1.44

Note: Use the positive zc in the formula for E.

Interpreting a Confidence IntervalExample 1.The interval we found is 10.45 < μ < 13.55With 95% confidence, we can say that the population mean is between 10.45 and 13.55.

Example 2.The interval we found is 9.66 < μ < 14.34With 95% confidence, we can say that the population mean is between 9.66 and 14.34.

Example 3.The interval we found is 0.552 < p < 0.648With 95% confidence, we can say that the population proportion is between 55.2% and 64.8%.

Comparing confidence intervals for a level of 90%, 95%, and 99%

n = 40xbar = 12σ = 5

For the 90% CI, E ≈ 1.30 and the interval is 10.70 < μ < 13.30

For the 95% CI, E ≈ 1.55 and the interval is 10.45 < μ < 13.55

For the 99% CI, E ≈ 2.04 and the interval is 9.96 < μ < 14.04

As the confidence level increases, the interval width increases. We have greater confidence, but less precision in estimating μ.

Have a great week!